1. Field of the Invention
This invention relates generally to the field of optical microlithography, and particularly to the use of triangle convolution to a sector-based OPC engine. More particularly, the invention relates to the use of triangle convolution on unbounded sectors, which inherently regularize the convolution of an intermediate-range, long-range, or infinitely extending kernel without spatially truncating the kernel, nor requiring excessive ROI size.
2. Description of Related Art
The optical microlithography process in semiconductor fabrication, also known as the photolithography process, consists of duplicating desired circuit patterns as best as possible onto a semiconductor wafer. The desired circuit patterns are typically represented as opaque and transparent regions on a template commonly called a photomask. In optical microlithography, patterns on the photomask template are projected onto photoresist-coated wafers by way of optical imaging through an exposure system.
Aerial image simulators, which compute the images generated by optical projection systems, have proven to be a valuable tool to analyze and improve the state-of-the-art in optical lithography for integrated circuit fabrications. These simulations have found application in advanced mask design, such as phase shifting mask (PSM) design, optical proximity correction (OPC), and in the design of projection optics. Modeling aerial images is a crucial component of semiconductor manufacturing. Since present lithographic tools employ partially coherent illumination, such modeling is computationally intensive for all but elementary patterns. The aerial image produced by the mask, i.e., the light intensity in an optical projection system's image plane, is a critically important quantity in microlithography for governing how well a developed photoresist structure replicates a mask design.
Mathematical difficulties arise in optical proximity correction calculations that, in part, can be attributed to divergence issues when attempting to analytically express imaging kernels over the unbound sectors into which mask polygons are conventionally decomposed. Attempts to account for incoherent flare are also subject to these analytical restrictions.
Accuracy is of critical importance in the computation of calibrated optical or resist models. The accuracy in the simulation of wafer shapes is necessary for a better understanding and evaluation of the correction methodologies. Through analytical processes, fidelity in the wafer shapes to the “as intended” shapes may ultimately achieve better correction of the mask shapes. An increase in yield during chip manufacturing is a direct consequence of achieving this accuracy.
The present invention introduces the utilization of triangle convolution to a sector-based optical proximity correction engine. Using triangles in the mathematical model to eliminate the difficulties of unbounded corners allows for integrals that accommodate intermediate-range and long-range effects.
In the prior art, the following mathematical treatment in the optical proximity correction engine is commonly used. These approaches are all, in one form or another, related to the Sum of Coherent Source (SOCS) method, which is an algorithm for efficient calculation of the bilinear transform.
Sum of Coherent Source (SOCS) Method
The image intensity is given by the partially coherent Hopkin's equation (a bilinear transform):I0({right arrow over (r)})=∫∫∫∫d{right arrow over (r)}′dr″h({right arrow over (r)}−{right arrow over (r)}′)h* ({right arrow over (r)}−{right arrow over (r)}″)j({right arrow over (r)}′−{right arrow over (r)}″)m({right arrow over (r)}′)m*({right arrow over (r)}″),                where,                    h is the lens point spread function (PSF);            j is the coherence;            m is the mask; and            I0 is the aerial image.                        
By using the SOCS technique, an optimal n-th order coherent approximation to the partially coherent Hopkin's equation can be expressed as             I      0        ⁡          (              r        →            )        ≅            ∑              k        =        1            n        ⁢                  ⁢                  λ        k            ⁢                                                            (                              m                ⊗                                  ϕ                  k                                            )                        ⁢                          (              x              )                                                2            
Where λk,φk({right arrow over (r)}) represents the eigenvalues and eigenvectors derived from the Mercer expansion of:             W      ⁡              (                                            r              →                        ′                    ,                      r            ″                          )              =                            h          ⁡                      (                                          r                →                            ′                        )                          ⁢                              h            *                    ⁡                      (                          r              ″                        )                          ⁢                  j          ⁡                      (                                                            r                  →                                ′                            -                              r                ″                                      )                              =                        ∑                      k            =            1                    ∞                ⁢                                  ⁢                              λ            k                    ⁢                                    ϕ              k                        ⁡                          (                                                r                  →                                ′                            )                                ⁢                                    ϕ              k                        ⁡                          (                                                r                  →                                ″                            )                                            ,which suggests that a partially coherent imaging problem can be optimally approximated by a finite summation of coherent imaging, such as linear convolution.SOCS with Pupil Phase Error
The above calculation assumes an ideal imaging system. However, when lens aberration is present, such as pupil phase error and apodization, one must include the pupil function:h({right arrow over (r)})=∫∫P({right arrow over (ρ)})exp(i W({right arrow over (ρ)}))exp(i2π{right arrow over (r)}·{right arrow over (ρ)})d2{right arrow over (ρ)}                where,                    P({right arrow over (ρ)}) is the pupil transmission function; and            W({right arrow over (ρ)}) is the pupil phase function, which contains both aberration and flare information.                        
Because of the possible higher spatial frequency in the wavefront function, h({right arrow over (r)}) will have a larger spatial extent. In this case, the number of eigenvalues and eigenvectors required will be higher than the ideal system. Hence, the kernel support area is extended to take account of the contribution from a distance greater than λ/NA. However, the basic mathematical structure and algorithm remains the same.
Physical Model of Flare
Flare is generally described as the image component generated by high frequency phase “ripples” in the wavefront. Flare thus arises when light is forward scattered by appreciable angles due to phase irregularities in a lens. Such irregularities are often neglected for three reasons. First, the wavefront data is sometimes taken with a low-resolution interferometer, and moreover may be reconstructed using an algorithm of even lower resolution. Second, even when the power spectrum of the wavefront is known or inferred, it is not possible to include the effect of high frequency wavefront components on an image integral that is truncated at a short ROI distance, causing most of the scattered light to be neglected. Last, it is not straightforward to include these terms in the calculated image. The present invention addresses these problems.
Bearing in mind the problems and deficiencies of the prior art, it is therefore an object of the present invention to provide a method for calculating intermediate and long-range image contributions from mask polygons.
It is another object of the present invention to apply triangle convolution techniques to a sector-based OPC engine.
It is a further object of the present invention to provide a method to calculate incoherent flare.
It is yet another object of the present invention to reduce the time needed for numerical integration for intermediate range calculation.
Another object of the present invention is to provide an analytical solution for the triangle convolution on a power law incoherent kernel.
A further object of the present invention is to present a programmable method to divide long-range and intermediate-range calculations by truncating mask but not kernel.
Another object of the present invention is to provide a method to account for the image component generated by high frequency phase “ripples” in the wavefront.
It is yet another object of the present invention to provide a method to calculate the effects of phase irregularities in a lens.
Another object of the present invention is to provide a method that includes the effect of high frequency wavefront components on an image integral that is truncated at a short ROI distance.
Still other objects and advantages of the invention will in part be obvious and will in part be apparent from the specification.